Outer-Inner Approximation Projection Methods for Multivalued Variational Inequalities

Abstract

In this paper, we present new projection methods for solving multivalued variational inequalities on a given nonlinear convex feasible domain. The first one is an extension of the extragradient method to multivalued variational inequalities under the asymptotic optimality condition, but it must satisfy certain Lipschitz continuity conditions. To avoid this requirement, we propose linesearch procedures commonly used in variational inequalities to obtain an approximation linesearch method for solving multivalued variational inequalities. Next, basing on a family of nonempty closed convex subsets of \(\mathcal R^{n}\) and linesearch techniques, we give inner approximation projection algorithms for solving multivalued variational inequalities and the convergence of the algorithms is established under few assumptions.